Metabelian SL(n,C) representations of knot groups

TL;DR

This paper classifies irreducible metabelian SL(n,C) representations of knot groups, showing finiteness and conjugacy properties linked to the Alexander polynomial, extending known results from SL(2,C) to higher ranks.

Contribution

It provides a classification and finiteness results for irreducible metabelian SL(n,C) representations of knot groups, generalizing previous SL(2,C) findings to higher rank cases.

Findings

Finiteness of conjugacy classes when the homology of the n-fold branched cover is finite.

All such representations are conjugate to unitary representations in the finite case.

A formula for the number of conjugacy classes in terms of the Alexander polynomial.

Abstract

We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2,C) representations of knot groups. Finally we deduce the existence irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial.